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The Geometric series formula is derived from the basic principle of geometric progression, where each term is a fixed multiple of the previous term. Learn more about the geometric series formula by reading below.
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Contents
Geometric series formula
A geometric series is a sequence of numbers in which each term is obtained by multiplying the preceding term by a fixed constant factor called the common ratio (r). The general form of a geometric series is:
a + ar + ar^2 + ar^3 + ar^4 + … = ∑(n=0 to infinity) ar^n
where a is the first term of the series, r is the common ratio, and n is the number of terms in the series.
The sum of a finite geometric series can be calculated using the following formula:
S_n = a(1-r^n) / (1-r)
where S_n is the sum of the first n terms of the series. This formula can be derived by multiplying the geometric series by the common ratio and subtracting the two series, resulting in a telescoping series that can be simplified to the above equation.
If the value of r is greater than 1, the geometric series will diverge to infinity because each term grows larger than the preceding term. If the value of r is between -1 and 1, the geometric series will converge to a finite sum because each term gets smaller and smaller as n increases.
If r is equal to 1, the series reduces to an arithmetic series with a constant difference between terms. In this case, the sum of the series can be calculated using the formula:
S_n = n(a + l) / 2
where l is the last term in the series.
The geometric series formula has many applications in mathematics, science, and finance. It can be used to model population growth, compound interest, and other phenomena that exhibit exponential behavior. It is also used in calculus to calculate infinite series and in physics to model the behavior of waves and other periodic phenomena.
In summary, the geometric series formula is a powerful tool for calculating the sum of a series of numbers that are related by a constant ratio. It provides a concise and elegant solution to a wide range of mathematical problems and is an essential tool for any student or practitioner of mathematics, science, or finance.
What is the formula for finding geometric series?
The formula for finding the sum of a geometric series is based on the geometric sequence, which is a sequence of numbers where each term is the product of the previous term and a fixed number called the common ratio. The formula for finding the sum of a finite geometric series is given by:
S_n = a(1 – r^n) / (1 – r)
where S_n is the sum of the first n terms of the geometric series, a is the first term of the series, r is the common ratio of the series, and n is the number of terms in the series.
To use this formula, we need to know the value of the first term (a), the common ratio (r), and the number of terms (n) in the series. Once we have these values, we can plug them into the formula and simplify to find the sum of the series.
For example, let’s say we have the geometric series:
2, 4, 8, 16, 32
To find the sum of the first 4 terms of this series, we would use the formula:
S_4 = 2(1 – 2^4) / (1 – 2)
Simplifying this expression, we get:
S_4 = 2(1 – 16) / -1
S_4 = -30
So the sum of the first 4 terms of the series is -30.
It’s important to note that the formula for finding the sum of an infinite geometric series is slightly different and is given by:
S = a / (1 – r)
where S is the sum of the infinite series, a is the first term of the series, and r is the common ratio of the series. However, this formula only works if the series converges, which means that the absolute value of the common ratio is less than 1.
In summary, the formula for finding the sum of a finite geometric series is S_n = a(1 – r^n) / (1 – r), where a is the first term of the series, r is the common ratio of the series, and n is the number of terms in the series. To find the sum of an infinite geometric series, the formula is S = a / (1 – r), but only if the series converges.
What is the rule of geometric series?
The rule of geometric series is a mathematical formula that allows us to find the sum of an infinite geometric sequence or a finite geometric sequence.
The formula for the sum of an infinite geometric series is:
S = a / (1 – r)
where S represents the sum of the series, a is the first term, and r is the common ratio between consecutive terms.
If the value of r is between -1 and 1, then the series converges and has a finite sum. If the value of r is outside of this range, the series diverges, and the sum is infinite.
For a finite geometric series, the formula for the sum is:
S = a ((1 – r^n) / (1 – r))
where n is the number of terms in the series.
It’s important to note that the formula assumes that the ratio r remains constant throughout the entire series. If the ratio changes, the formula will no longer be valid.
In summary, the rule of geometric series provides a formula for calculating the sum of either an infinite or finite geometric sequence.
What is geometric series with example?
In mathematics, a geometric series is a sequence of numbers in which each term is obtained by multiplying the previous term by a fixed non-zero number called the common ratio. The general form of a geometric series is given by:
a + ar + ar^2 + ar^3 + … = ∑(ar^n) from n = 0 to infinity
where a is the first term and r is the common ratio.
For example, let us consider the geometric series:
3 + 6 + 12 + 24 + 48 + …
To determine the common ratio, we divide any term by its preceding term:
6/3 = 2
12/6 = 2
24/12 = 2
48/24 = 2
Therefore, the common ratio r = 2.
Now we can use the formula for the sum of a geometric series:
S_n = a(1 – r^n) / (1 – r)
where S_n is the sum of the first n terms of the series.
Let’s find the sum of the first 5 terms of the series:
a = 3
r = 2
n = 5
S_5 = 3(1 – 2^5) / (1 – 2) = 3(-31) / (-1) = 93
Therefore, the sum of the first 5 terms of the series is 93.
Another example of a geometric series is:
1/2 + 1/4 + 1/8 + 1/16 + …
In this case, the first term is a = 1/2 and the common ratio is r = 1/2. Using the formula for the sum of a geometric series, we can find that the sum of the first 4 terms of the series is:
a = 1/2
r = 1/2
n = 4
S_4 = (1/2)(1 – (1/2)^4) / (1 – 1/2) = 1 – 1/16 = 15/16
Therefore, the sum of the first 4 terms of the series is 15/16.
Geometric series are used in various areas of mathematics and science, such as finance, physics, and engineering.
How to use the geometric series formula?
The formula for the sum of a geometric series is:
S = a(1 – r^n)/(1 – r)
where:
- S is the sum of the geometric series
- a is the first term in the series
- r is the common ratio between the terms in the series
- n is the number of terms in the series
To use the geometric series formula, follow these steps:
Step 1: Identify the first term (a), the common ratio (r), and the number of terms (n) in the series.
Step 2: Plug the values of a, r, and n into the formula:
S = a(1 – r^n)/(1 – r)
Step 3: Simplify the expression by expanding the terms and combining like terms.
Step 4: Evaluate the expression to find the sum of the geometric series.
Here’s an example of using the geometric series formula:
Find the sum of the geometric series 2, 4, 8, 16, 32, 64.
Step 1: Identify a, r, and n.
- a = 2 (the first term)
- r = 2 (the common ratio)
- n = 6 (the number of terms)
Step 2: Plug in the values:
S = 2(1 – 2^6)/(1 – 2)
Step 3: Simplify the expression:
S = 2(-63)/(-1)
S = 126
Step 4: Evaluate the expression:
The sum of the geometric series is 126.
Note that in this example, the formula works because the series has a finite number of terms. If the series were infinite, the sum would approach infinity, but not have a finite value. In such cases, we use a different formula for the sum of an infinite geometric series.
Is geometric series a sequence?
Geometric series can be considered both a sequence and a series, depending on the context in which it is used.
A sequence is a list of numbers arranged in a specific order, whereas a series is the sum of the terms of a sequence. In the case of a geometric sequence, the terms are generated by multiplying each previous term by a constant ratio. For example, the sequence 1, 2, 4, 8, 16 is a geometric sequence with a common ratio of 2, since each term is twice the previous term.
However, if we add up the terms of the sequence, we get a series. The sum of a finite geometric series can be calculated using the formula:
S_n = a(1 – r^n)/(1 – r)
where S_n is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms.
For example, let’s consider the geometric sequence 1, 2, 4, 8, 16. If we want to find the sum of the first 4 terms of the sequence, we can use the formula above with a=1, r=2, and n=4:
S_4 = 1(1 – 2^4)/(1 – 2) = 1(-15)/(-1) = 15
So the sum of the first 4 terms of the sequence is 15. Therefore, we can say that the geometric sequence 1, 2, 4, 8, 16 is also a geometric series with a sum of 15.
In summary, a geometric sequence is a sequence of numbers that follow a specific pattern, while a geometric series is the sum of the terms of a geometric sequence. Therefore, geometric series can be considered both a sequence and a series depending on the context in which it is used.
Geometric series formula – FAQ
1. What is the geometric series formula?
The geometric series formula is a mathematical equation used to calculate the sum of an infinite geometric series or a finite geometric series.
2. What is a geometric series?
A geometric series is a sequence of numbers in which each term is obtained by multiplying the previous term by a fixed constant called the common ratio.
3. What is the formula for finding the sum of a finite geometric series?
The formula for finding the sum of a finite geometric series is Sn = a(1 – r^n) / (1 – r), where a is the first term, r is the common ratio, and n is the number of terms.
4. What is the formula for finding the sum of an infinite geometric series?
The formula for finding the sum of an infinite geometric series is S = a / (1 – r), where a is the first term and r is the common ratio.
5. What is the common ratio in a geometric series?
The common ratio in a geometric series is the ratio between any two consecutive terms in the sequence.
6. What is the first term in a geometric series?
The first term in a geometric series is the initial term of the sequence.
7. Can a geometric series have a negative common ratio?
Yes, a geometric series can have a negative common ratio.
8. What is the significance of the geometric series formula in mathematics?
The geometric series formula is significant in mathematics because it allows us to calculate the sum of an infinite series and to analyze the behavior of geometric sequences.
9. How is the geometric series formula used in finance?
The geometric series formula is used in finance to calculate the present value and future value of annuities, bonds, and other financial instruments.
10. How is the geometric series formula used in physics?
The geometric series formula is used in physics to model exponential decay and growth phenomena, such as radioactive decay and population growth.
11. What is the relationship between geometric series and exponential functions?
Geometric series and exponential functions are closely related because a geometric series can be expressed as an exponential function, and vice versa.
12. Can the geometric series formula be applied to non-numeric sequences?
The geometric series formula is generally applied to numeric sequences, but it can be extended to other types of sequences, such as geometric shapes or patterns.
13. How does the value of the common ratio affect the behavior of a geometric series?
The value of the common ratio affects the behavior of a geometric series because it determines whether the series converges to a finite value or diverges to infinity.
14. What is the difference between an arithmetic series and a geometric series?
An arithmetic series is a sequence in which each term is obtained by adding a fixed constant to the previous term, while a geometric series is a sequence in which each term is obtained by multiplying the previous term by a fixed constant.
15. How is the sum of a geometric series related to its terms?
The sum of a geometric series is related to its terms because it is the sum of the product of each term with the common ratio raised to the power of its position in the sequence.
16. What is the role of the geometric series formula in calculus?
The geometric series formula is used in calculus to evaluate certain types of integrals and to derive important mathematical concepts, such as the Taylor series expansion.
17. Can the geometric series formula be used for non-integer values of the common ratio?
Yes, the geometric series formula can be used for non-integer values of the common ratio, as long as the series converges.
18. What is the geometric series formula when the common ratio is greater than 1?
The geometric series formula is only valid when the absolute value of the common ratio (r) is less than 1. If the common ratio is greater than 1, the terms in the series will grow infinitely large, and the series will not converge.
19. Can the geometric series formula be used to find the sum of an infinite series?
Yes, the geometric series formula can be used to find the sum of an infinite series, as long as the series converges. When the number of terms (n) in the formula is replaced with infinity (∞), the formula becomes S = a/(1 – r), where S is the sum of the infinite series.
20. How is the geometric series formula related to compound interest?
The geometric series formula is closely related to the calculation of compound interest. If an investment earns a fixed interest rate (r) each period, the value of the investment after n periods can be calculated using the formula A = P(1 + r)^n, where A is the final amount, P is the initial amount, and (1 + r) is the common ratio between successive terms. If the interest is compounded continuously, the formula becomes A = Pe^(rt), where e is Euler’s number (approximately 2.718), and t is the time in years. This formula is an example of a continuous geometric series.
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